Nonlinear amplitude dynamics in flagellar beating

Abstract

The physical basis of flagellar and ciliary beating is a major problem in biology which is still far from completely understood. The fundamental cytoskeleton structure of cilia and flagella is the axoneme, a cylindrical array of microtubule doublets connected by passive cross-linkers and dynein motor proteins. The complex interplay of these elements leads to the generation of self-organized bending waves. Although many mathematical models have been proposed to understand this process, few attempts have been made to assess the role of dyneins on the nonlinear nature of the axoneme. Here, we investigate the nonlinear dynamics of flagella by considering an axonemal sliding control mechanism for dynein activity. This approach unveils the nonlinear selection of the oscillation amplitudes, which are typically either missed or prescribed in mathematical models. The explicit set of nonlinear equations are derived and solved numerically. Our analysis reveals the spatio-temporal dynamics of dynein populations and flagellum shape for different regimes of motor activity, medium viscosity and flagellum elasticity. Unstable modes saturate via the coupling of dynein kinetics and flagellum shape without the need of invoking a nonlinear axonemal response. Hence, our work reveals a novel mechanism for the saturation of unstable modes in axonemal beating.

The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella

“…The boundary conditions are non-local due to the accumulative dependence of sliding moments along the bundle, and generally unknown during the dynamics. This is becomes even more challenging when the bundle is driven via the molecular motor activity [20,68], which may explain the reason why numerical investigation of geometrically exact bundle systems are still lacking, for both active and passive cases [68]. The coarse-grained formulation brakes the contribution of the sliding filament moments for each segments simply as…”

The human sperm beats anisotropically and asymmetrically in 3D

“…Furthermore, the standing waves in the z-plane are unlikely to be achieved via molecular motor dynamics alone as this would require a fast, synchronised motor coordination over long flagellar distances, not yet reported in the literature. Indeed, molecular motor self-organisation is manifested through time-delayed oscillations along the flagellum [38,63,64] which leads to the generation of propagating waves, and therefore the most likely mechanism to govern the asymmetric travelling waves observed in the b-plane ( Fig. 8(a)) [1,2,5].…”

“…We employed principal component analysis (PCA) on the 3D flagellar shape at the comoving, X c (s, t), and comoving-rolling, X cr (s, t), frames of reference. For this, we generalised the previous PCA of scalar quantities [26,38,65,66] to three-dimensional vectorial functions, in which each spatial coordinate represents an spatiotemporal map in arclength and time, (s, t).…”

“…As such, beating symmetry is the cornerstone of any comparative study on flagellar undulation [24,29,32], especially so while assessing symmetry-breaking events, such as in sperm capacitation, hyperactivation and buckling phenomena [18,19,[32][33][34]. Theoretical and computational models equally, and ubiquitously, invoke symmetry, ranging from flagellar kinematic models to structural mechanics and even motor control hypothesis [5,10,12,26,[35][36][37][38][39].…”

Efficient implementation of elastohydrodynamics via integral operators

“…The potential applications for a fast and accurate filament modeling framework are numerous. There has long been interest in understanding the mechanics and regulation of sperm flagellar movement, in particular problems relating to: understanding the mechanical struc-ture and motor regulation [5,[12][13][14], investigating the response of the flagellar beat to its rheological environment [15][16][17], understanding the dynamics of sperm due to surrounding solid walls [18,19], and studying the effect of viscosity on sperm swimming [20]. For a detailed review surrounding the importance of the sperm flagellum see Gallagher et al [21].…”